The above analysis is versatile for computing the commutator dependence of Trotter error. Unfortunately, the resulting bound does not use prior knowledge of the initial state and will in particular be loose if the initial state lies within a low-energy subspace. On the other hand, recent work of Şahinoğlu and Somma proposed a Trotterization approach for simulating low-energy initial states but the commutativity of the Hamiltonian was ignored in their analysis [1]}. Here, we address this by simultaneously using commutativity of the Hamiltonian and prior knowledge of the initial state to improve the simulation of a class of interacting electrons. We obtain further improvement when the electronic Hamiltonian has sparse interactions. In the following, we introduce preliminaries about the second-quantization representation ([sec:prelimquantization]Section sec:prelimquantization) and the notion of fermionic seminorm ([sec:prelimseminorm]Section sec:prelimseminorm), on which our analysis will be based.
